082-Engineering-Mathematics-Anthony-Croft-Robert-Davison-Martin-Hargreaves-James-Flint-Edisi-5-2017

776 Chapter 24 The Fourier transform
g(–l)
f( l)= u( l)e – l
(a)
l
g(t – l) when t < 0
(b)
t
l
g(t – l) when t > 0
(c)
t
l
Figure24.13
Thefunctiong(t − λ)forvariousvalues
oft.
Whent<0
Whent < 0there isclearlyno overlap and itfollows that f ∗g= 0.
Whent0
Whent 0thereisoverlapforvaluesof λbetween0andt,thatiswhen0 λ t,
and hence
f∗g=
=
∫ t
0
∫ t
0
e −λ e −2(t−λ) dλ
e −λ e −2t e 2λ dλ
∫ t
= e −2t e λ dλ
0
= e −2t [e λ ] t 0
= e −2t (e t −1)
= e −t −e −2t
Finally the complete expression for the convolution is
(f ∗g)(t)=
This may alsobe writtenas
{ e −t −e −2t whent 0
0 whent<0
(f ∗g)(t) =u(t)(e −t −e −2t )
(b) UsingTable 24.1 the Fourier transforms of f andgare
F(ω) = 1
1+jω
G(ω) = 1
2+jω